Method for estimating compliance at points along a beam from bending measurements

Abstract

The inherent smoothing in bending stiffness measurement of wood boards as occurs in the machine grading of lumber can mask the effect of knots and other local characteristics affecting structural value. Improved estimates of local (pointwise) stiffness will be useful in decisions about further processing and use of a tested board. Measured compliance is reciprocally related to measured stiffness and is the convolution of local compliance and a “span function”. Span function is specific to the bending span configuration used and can change during measurement of a board. A general procedure for computing span function, which heretofore has been known only for simple bending spans, is disclosed. A Kalman filter uses this and other available information to optimally estimate local compliance from an observed relationship between local compliance and state variables of a state-space model. Method for linear algebraic determination of local compliance also depends on span functions and is disclosed.

1. A computer-implemented method of computing span function for a bending span used to measure compliance of an elongated beam at a measurement point on the beam and using the span function in the estimation of local compliance values at points along the beam, the method comprising the following steps:
 defining a compliance test function comprising a background compliance plus a compliance impulse of weight b at position x relative to the measurement point;  
 obtaining an expression of measured compliance as a function C m (b,x) of impulse weight b and position x;  
 computing the span function h(x) as being substantially equal to the partial derivative, of C m (b,x) with respect to the impulse weight b, evaluated at b=0; and  
 using a computer to implement an algorithm dependent on the span function h(x) in the estimation of local compliance values.  
 
   
   
     2. The method of  claim 1  for computing and using span function applied to a plurality of Ψ numbered bending spans giving a corresponding plurality of Ψ numbered compliance measurements at a corresponding plurality of Ψ numbered measurement points, not necessarily all distinct, along the beam, yielding a corresponding plurality of Ψ numbered span functions, and using the span functions and compliance measurements to obtain a least squares solution for an (N,1)-dimensional vector C with components representing N local compliance values at points spaced along the beam, by the following steps:
 forming a (Ψ,1)-dimensional vector C m  with numbered components comprised of the corresponding plurality of Ψ numbered compliance measurements;  
 forming a (Ψ,N)-dimensional rectangular matrix H having Ψ numbered rows, each row being a (1,N)-dimensional matrix in correspondence with a compliance measurement and a span function and comprised of entries computed from the span function, the entries positioned in the row so that the matrix product of the row and vector C is a scalar linear combination of components of vector C corresponding to a compliance measurement in C m ; and  
 computing C as a least squares solution to HC=C m , the solution being contained in a linear manifold having dimension N−r, r being the rank of matrix H.  
 
   
   
     3. The method of  claim 2  wherein the matrix H is augmented to matrix H a  and the measured compliance vector is augmented to vector C ma , giving a least squares solution C a  of the equation C m =H a C a , the solution C a  being in the linear manifold, a plurality of first components of C a  being a uniform value and a plurality of last components of C a  being a uniform value; whereby C a  makes physical sense as well as being in the linear manifold. 
   
   
     4. A computer-implemented method of obtaining a local compliance estimate at a point of estimation on an elongated beam, from a sequence of “m” measured compliance values at “m” measurement points spaced along the beam, each measured compliance value being obtained by applying a bending span to a length segment of the beam, the length segment including the point of estimation and having unknown local compliance values along its length, the length segment and measured compliance value being identified with a corresponding measurement point on the beam; thereby defining a sequence of corresponding “m” measured compliance values, “m” measurement points, “m” bending spans and “m” length segments; the method comprising the following steps:
 representing each measured compliance value minus an estimated mean value common to the measured compliance sequence as being the output from a state-space representation of a dynamic system, the state-space representation comprising a vector state equation and a scalar output equation, the state equation containing a state matrix, a state vector, and an input vector with at least one component being a white random noise source, the state equation describing how the state vector with component state variables changes from one measurement point to the next, the local compliance values from the corresponding length segment minus the common estimated mean value being represented by the state variables, the output equation having an output matrix specific to a corresponding bending span and having a measurement white random noise source independent of input vector noise, the output equation specifying for each measurement point the dynamic system output as a linear combination of the state variables plus measurement noise;  
 using a priori information to initialize a Kalman filter by initializing estimates of the state vector, input vector covariance matrix, measurement noise variance, and state vector covariance matrix;  
 applying the Kalman filter recursively to the sequence of “m” measured compliance values minus the common estimated mean value;  
 computing from the Kalman filter a sequence of “m” state vector estimates, one corresponding to each member of the measured compliance value sequence; and  
 obtaining the local compliance estimate at the point of estimation as the common estimated mean value plus a selected component from a selected state vector estimate in the sequence of “m” state vector estimates.  
 
   
   
     5. The method of  claim 4  wherein the coefficients of each output matrix are computed from a span function for the corresponding bending span, the span function computed according to the following steps:
 defining a compliance test function comprising a background compliance plus a compliance impulse of weight b at position x relative to the corresponding measurement point;  
 deriving an expression of measured compliance as a function C m (b,x) of impulse weight b and position x; and  
 computing the span function h(x) as being substantially equal to the partial derivative, of C m (b,x) with respect to the impulse weight b, evaluated at b=0.  
 
   
   
     6. The method of  claim 5  wherein the selected state vector estimate is the last one in the sequence of state vector estimates. 
   
   
     7. The method of  claim 5  wherein the selected state vector estimate is the last one in the sequence of state vector estimates for which the selected component has, in the corresponding output matrix, a coefficient magnitude exceeding a selected threshold value. 
   
   
     8. The method of  claim 5  applied to obtain a sequence of “n” local compliance estimates at corresponding “n” points of estimation spaced along the length of the beam, the sequence of “n” points of estimation beginning substantially at one end of the beam and ending substantially at the other end of the beam, the sequence of “n” local compliance estimates obtained from a sequence of “M” compliance measurements at a corresponding sequence of “M” measurement points on the beam and a corresponding computed sequence of “M” state vector estimates, the sequence of “M” measurement points being a coalesced grand sequence of measurement points from the “n” sequences of measurement points for the “n” points of estimation, the common estimated mean value being common to all members of the coalesced grand sequence. 
   
   
     9. The method of  claim 8  wherein, for each point of estimation, the selected state vector estimate is the last one in the sequence of “M” state vector estimates that has a component representing, at the point of estimation, the local compliance minus the common estimated mean value. 
   
   
     10. The method of  claim 8  wherein, for each point of estimation, the selected state vector estimate is the last one in the sequence of “M” state vector estimates that has a component representing, at the point of estimation, the local compliance minus the common estimated mean value and also has, in the corresponding output matrix, a coefficient magnitude exceeding a selected threshold value. 
   
   
     11. The method of  claim 8  wherein the “M” compliance measurements minus the common estimated mean value and minus the random measurement noise are modeled as coming from an autoregressive moving average (ARMA) random process, the autoregressive coefficients and input noise comprising the autoregressive part of the ARMA model and yielding, for the state-space dynamic system model, state variables as an autoregressive random process that models the sequence of local compliance values minus the common estimated mean value, the autoregressive coefficients appearing in the state matrix, the moving average coefficients of the ARMA model for each compliance measurement being weighting coefficients in the output matrix of the state-space model, the moving average coefficients changing in correspondence with bending span changes, and wherein the initial state vector covariance, state matrix, and input vector covariance satisfy substantially a discrete Lyapunov equation. 
   
   
     12. The method of  claim 11  wherein additionally, autoregressive coefficients of the ARMA model are obtained as a priori information from compliance measurements, estimation of autocorrelations of measured compliance, and solution of a system of equations relating autocorrelations and autoregressive coefficients. 
   
   
     13. The method of  claim 11  wherein the autoregressive part of the model is simplified to include at most two autoregressive coefficients. 
   
   
     14. The method of  claim 13  wherein one autoregressive coefficient is used with a first input random white noise source and a second autoregressive coefficient is used with a second input random white noise source independent of the first in a parallel branch, the two branches being used in the autoregressive part of the ARMA model. 
   
   
     15. The method of  claim 8  wherein, additionally, observed systematic measurement noise is modeled, included in the output of the state-space representation of the dynamic system, and estimated from a component of the state vector corresponding to each measured compliance value; whereby the estimated systematic noise may be used as an indicator of performance of the measuring apparatus. 
   
   
     16. The method of  claim 8  wherein additionally a measure of estimation quality is computed for each local compliance estimate. 
   
   
     17. The method of  claim 16  wherein additionally estimated local E is computed as a corrected reciprocal of estimated local compliance, and a measure of estimation quality is computed for each estimate of local E as a corrected measure of estimation quality for local compliance. 
   
   
     18. The method of  claim 8  wherein a plurality of measurement sequences are arranged as a vector measurement sequence, each component of the vector measurement sequence corresponding to a different sequence of bending spans but corresponding to substantially the same measurement points as the other components of the vector measurement sequence, the state-space dynamic system model being the same as in  claim 8  except for additional rows in the output matrix for the output equation, the output being a vector output, the Kalman filter operating recursively on the vector measurement sequence minus an estimated mean vector common to each member of the vector measurement sequence, the Kalman filter providing for each measurement point an optimal estimate of the state vector, the local compliance estimates being determined through their correspondence with selected components of selected estimated state vectors. 
   
   
     19. A computer-implemented calibration method in a machine for measuring modulus of elasticity of an elongated beam, the machine applying a sequence of bending spans to a corresponding sequence of length segments at a corresponding sequence of measurement points along the beam, each bending span having defined support specifications, wherein the computer is programmed to apply a sequence of calibration factors in correspondence with the bending spans, each factor specific to its corresponding bending span and determined as the factor, which, when applied during the measurement of a beam having a uniform modulus of elasticity, causes the measured modulus of elasticity to be that uniform value for each bending span in the sequence; whereby machines, having a plurality of bending spans and designed to use just one calibration factor, can be made to give accurate readings for each bending span even if support conditions deviate away from the defined support specifications. 
   
   
     20. The calibration method of  claim 19  wherein the computation of each calibration factor specific to its corresponding bending span is substantially equivalent to adjusting the factor so that the coefficients in an output matrix sum to one, the output matrix being the output matrix of a state-space representation of a dynamic system modeling the reciprocal of the modulus of elasticity measurement minus a common estimated mean reciprocal modulus of elasticity value as the output of the dynamic system.